Chapter 2 Discrete-Time Market Model
The single-step model considered in Chapter1 is extended to a discrete-time model with N + 1 time instants t = 0, 1, ... , N. A basic limitation of the one-step model is that it does not allow for trading until the end of the first time period is reached, while the multistep model allows for multiple portfolio re-allocations over time. The Cox-Ross-Rubinstein (CRR) model, or binomial model, is considered as an example whose importance also lies with its computer implementability.
2.1 Discrete-Time Compounding ...... 51 2.2 Arbitrage and Self-Financing Portfolios ...... 54 2.3 Contingent Claims ...... 60 2.4 Martingales and Conditional Expectations ...... 65 2.5 Market Completeness and Risk-Neutral Measures 71 2.6 The Cox-Ross-Rubinstein (CRR) Market Model . . 73 Exercises ...... 78
2.1 Discrete-Time Compounding
Investment plan
We invest an amount m each year in an investment plan that carries a constant interest rate r. At the end of the N-th year, the value of the amount m invested at the beginning of year k = 1, 2, ... , N has turned into (1 + r)N−k+1m and the value of the plan at the end of the N-th year becomes
N X N−k+1 AN : = m (1 + r) (2.1) k=1
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N X = m (1 + r)k k=1 (1 + r)N − 1 = m(1 + r) , r hence the duration N of the plan satisfies
1 rAN N + 1 = log 1 + r + . log(1 + r) m
Loan repayment
At time t = 0 one borrows an amount A1 := A over a period of N years at the constant interest rate r per year. Proposition 2.1. Constant repayment. Assuming that the loan is completely repaid at the beginning of year N + 1, the amount m refunded every year is given by
r(1 + r)N A r m = = A. (2.2) (1 + r)N − 1 1 − (1 + r)−N
Proof. Denoting by Ak the amount owed by the borrower at the beginning o of year n k = 1, 2, ... , N with A1 = A, the amount m refunded at the end of the first year can be decomposed as
m = rA1 + (m − rA1), into rA1 paid in interest and m − rA1 in principal repayment, i.e. there remains
A2 = A1 − (m − rA1) = (1 + r)A1 − m, to be refunded. Similarly, the amount m refunded at the end of the second year can be decomposed as
m = rA2 + (m − rA2), into rA2 paid in interest and m − rA2 in principal repayment, i.e. there remains
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= (1 + r)((1 + r)A1 − m) − m 2 = (1 + r) A1 − m − (1 + r)m to be refunded. After repeating the argument we find that at the beginning of year k there remains
k−1 k−2 Ak = (1 + r) A1 − m − (1 + r)m − · · · − (1 + r) m k−2 k−1 X i = (1 + r) A1 − m (1 + r) i=0 1 − (1 + r)k−1 = (1 + r)k−1A + m 1 r to be refunded, i.e.
m − (1 + r)k−1(m − rA) A = , k = 1, 2, ... , N. (2.3) k r
We also note that the repayment at the end of year k can be decomposed as
m = rAk + (m − rAk), with k−1 rAk = m + (1 + r) (rA1 − m) in interest repayment, and
k−1 m − rAk = (1 + r) (m − rA1) in principal repayment. At the beginning of year N + 1, the loan should be completely repaid, hence AN+1 = 0, which reads
1 − (1 + r)N (1 + r)N A + m = 0, r and yields (2.2). We also have
A 1 − (1 + r)−N = . (2.4) m r and
1 m log(1 − rA/m) N = log = − . log(1 + r) m − rA log(1 + r)
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Remark: One needs m > rA in order for N to be finite.
The next proposition is a direct consequence of (2.2) and (2.3). Proposition 2.2. The k-th interest repayment can be written as
N−k+1 1 X −l rAk = m 1 − = mr (1 + r) , (1 + r)N−k+1 l=1 and the k-th principal repayment is
m m − rAk = , k = 1, 2, ... , N. (1 + r)N−k+1
Note that the sum of discounted payments at the rate r is
N X m 1 − (1 + r)−N = m = A. (1 + r)l r l=1 In particular, the first interest repayment satisfies
N X 1 −N rA = rA1 = mr = m 1 − (1 + r) , (1 + r)l l=1 and the first principal repayment is m m − rA = . (1 + r)N
2.2 Arbitrage and Self-Financing Portfolios
Stochastic processes
A stochastic process on a probability space (Ω, F, P) is a family (Xt)t∈T of random variables Xt : Ω −→ R indexed by a set T . Examples include: • the one-step (or two-instant) model: T = {0, 1},
• the discrete-time model with finite horizon: T = {0, 1, ... , N},
• the discrete-time model with infinite horizon: T = N,
• the continuous-time model: T = R+.
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For real-world examples of stochastic processes one can mention:
• the time evolution of a risky asset, e.g. Xt represents the price of the asset at time t ∈ T .
• the time evolution of a physical parameter - for example, Xt represents a temperature observed at time t ∈ T . In this chapter, we focus on the finite horizon discrete-time model with T = {0, 1, ... , N}.
0 1 2 3 N
Asset price modeling
The prices at time t = 0 of d + 1 assets numbered 0, 1, ... , d are denoted by the random vector (0) (1) (d) S0 = S0 , S0 , ... , S0 in Rd+1. Similarly, the values at time t = 1, 2, ... , N of assets no 0, 1, ... , d are denoted by the random vector
(0) (1) (d) St = St , St , ... , St
S on Ω, which forms a stochastic process t t=0,1,...,N .
In the sequel we assume that asset no 0 is a riskless asset (of savings account type) yielding an interest rate r, i.e. we have
(0) t (0) St = (1 + r) S0 , t = 0, 1, ... , N.
Portfolio strategies
Definition 2.3. A portfolio strategy is a stochastic process ξ ⊂ t t=1,2,...,N d+1 (k) o R where ξt denotes the (possibly fractional) quantity of asset n k held in the portfolio over the time interval (t − 1, t], t = 1, 2, ... , N. Note that the portfolio allocation
(0) (1) (d) ξt = ξt , ξt , ... , ξt is decided at time t − 1 and remains constant over the interval (t − 1, t] while (k) (k) the stock price changes from St−1 to St over this time interval.
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ξ1 ξ2 ξ3 ξN
0 1 2 3 N
In other words, the quantity (k) (k) ξt St−1 represents the amount invested in asset no k at the beginning of the time interval (t − 1, t], and (k) (k) ξt St represents the value of this investment at the end of the time interval (t − 1, t], t = 1, 2, ... , N.
Self-financing portfolio strategies
The opening price of the portfolio at the beginning of the time interval (t − 1, t] is d X (k) (k) ξt • St−1 = ξt St−1, k=0 when the market “opens” at time t − 1. When the market “closes”at the end of the time interval (t − 1, t], it takes the closing value
d X (k) (k) ξt • St = ξt St , (2.5) k=0 t = 1, 2, ... , N. After the new portfolio allocation ξt+1 is designed we get the new portfolio opening price
d X (k) (k) • ξt+1 St = ξt+1St , (2.6) k=0 at the beginning of the next trading session (t, t + 1], t = 0, 1, ... , N − 1.
Note that here, the stock price St is assumed to remain constant “overnight”, i.e. from the end of (t − 1, t] to the beginning of (t, t + 1], t = 1, 2, ... , N − 1.
In case (2.5) coincides with (2.6) for t = 0, 1, ... , N − 1 we say that the ξ self-financing portfolio strategy t t=1,2,...,N is . A non self-financing portfolio could be either bleeding money, or burning cash, for no good reason. Definition 2.4. A portfolio strategy ξ is said to be t t=1,2,...,N self-financing if
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ξt • St = ξt+1 • St, t = 1, 2, ... , N − 1, (2.7) i.e. d d X (k) (k) X (k) (k) ξt St = ξt+1St , t = 1, 2, ... , N − 1. k=0 k=0 | {z } | {z } Closing value Opening price The meaning of the self-financing condition (2.7) is simply that one cannot take any money in or out of the portfolio during the “overnight” transition period at time t. In other words, at the beginning of the new trading session (t, t + 1] one should re-invest the totality of the portfolio value obtained at the end of the interval (t − 1, t].
The next figure is an illustration of the self-financing condition.
Portfolio value ξtSt−1 ξtSt = ξt+1St ξt+1St+1 Asset value St−1 St St St+1
Time scale t − 1 t t t + 1 Portfolio allocation ξt ξt ξt+1 ξt+1
“Morning” “Evening” “Next morning” “Next evening” (Opening) (Closing) (Opening) (Closing) Fig. 2.1: Illustration of the self-financing condition (2.7).
By (2.5) and (2.6) the self-financing condition (2.7) can be rewritten as
d d X (k) (k) X (k) (k) ξt St = ξt+1St , t = 0, 1, ... , N − 1, k=0 k=0 or d X (k) (k) (k) ξt+1 − ξt St = 0, t = 0, 1, ... , N − 1. k=0 ξ Note that any portfolio strategy t t=1,2,...,N which is constant over time, i.e. ξt = ξt+1, t = 1, 2, ... , N − 1, is self-financing by construction.
Here, portfolio re-allocation happens “overnight”, during which time the global portfolio value remains the same due to the self-financing condition. The portfolio allocation ξt remains the same throughout the day, however the portfolio value changes from morning to evening due to a change in the stock price. Also, ξ0 is not defined and its value is actually not needed in this
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(0) (1) In case d = 1 we are only trading d + 1 = 2 assets St = St , St and (0) (1) the portfolio allocation reads ξt = ξt , ξt . In this case, the self-financing condition means that: (1) • In the event of an increase in the stock position ξt , the corresponding (1) (1) (1) cost of purchase ξt+1 − ξt St > 0 has to be deducted from the savings (0) (0) account value ξt St , which becomes updated as
(0) (0) (0) (0) (1) (1) (1) ξt+1St = ξt St − ξt+1 − ξt St , recovering (2.7). (1) • In the event of a decrease in the stock position ξt , the corresponding sale (1) (1) (1) profit ξt − ξt+1 St > 0 has to be added to the savings account value (0) (0) ξt St , which becomes updated as
(0) (0) (0) (0) (1) (1) (1) ξt+1St = ξt St + ξt − ξt+1 St , recovering (2.7). Clearly, the chosen unit of time may not be the day and it can be replaced by weeks, hours, minutes, or fractions of seconds in high-frequency trading.
Portfolio value
Definition 2.5. The portfolio opening prices at times t = 0, 1, ... , N − 1 are defined as
d X (k) (k) • Vt := ξt+1 St = ξt+1St , t = 0, 1, ... , N − 1. k=0
Under the self-financing condition (2.7), the portfolio closing values Vt at time t = 1, 2, ... , N rewrite as
d X (k) (k) Vt = ξt • St = ξt St , t = 1, 2, ... , N, (2.8) k=0 as summarized in the following table.
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V0 V1 V2 ······ VN−1 VN Opening price ξ1 • S0 ξ2 • S1 ξ3 • S2 ······ ξN • SN−1 N.A. Closing value N.A. ξ1 • S1 ξ2 • S2 ······ ξN−1 • SN−1 ξN • SN
Table 2.1: Self-financing portfolio value process.
Discounting
My portfolio St grew by b = 5% this year. My portfolio St grew by b = 5% this year. The risk-free or inflation rate is r = 10%. Q: Did I achieve a positive return? Q: Did I achieve a positive return? A: A:
(a) Scenario A. (b) Scenario B.
Fig. 2.2: Why apply discounting?
Definition 2.6. Let (0) (1) (d) Xt := Set , Set , ... , Set ) denote the vector of discounted asset prices, defined as:
(i) 1 (i) Se = S , i = 0, 1, ... , d, t = 0, 1, ... , N. t (1 + r)t t
(a) Without inflation adjustment. (b) With inflation adjustment.
Fig. 2.3: Are oil prices higher in 2019 compared to 2005?
We can also write
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1 Xt := St, t = 0, 1, ... , N. (1 + r)t
The discounted value at time 0 of the portfolio is defined by 1 Vet = Vt, t = 0, 1, ... , N. (1 + r)t
For t = 1, 2, ... , N we have 1 Vet = ξt • St (1 + r)t d 1 X k k = ξ( )S( ) (1 + r)t t t k=0 d X (k) (k) = ξt Set k=0 = ξt • Xt, while for t = 0 we get Ve0 = ξ1 • X0 = ξ1 • S0. The effect of discounting from time t to time 0 is to divide prices by (1 + r)t, making all prices comparable at time 0.
Arbitrage
The definition of arbitrage in discrete time follows the lines of its analog in the one-step model. Definition 2.7. A portfolio strategy ξ constitutes an arbitrage t t=1,2,...,N opportunity if all three following conditions are satisfied:
i) V0 6 0 at time t = 0, [Start from a zero-cost portfolio, or with a debt.]
ii) VN > 0 at time t = N, [Finish with a nonnegative amount.]
iii) P(VN > 0) > 0 at time t = N.[Profit is made with nonzero probability.]
2.3 Contingent Claims
Recall that from Definition 1.8, a contingent claim is given by the nonneg- ative random payoff C of an option contract at maturity time t = N. For example, in the case of the European call option of Definition2, the payoff C (i) + is given by C = SN − K where K is called the strike (or exercise) price of the option, while in the case of the European put option of Definition1 60 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance
(i)+ we have C = K − SN .
The list given below is somewhat restrictive and there exists many more option types, with new ones appearing constantly on the markets.
Physical delivery vs cash settlement
(i) + The cash settlement realized through the payoff C = SN − K can be replaced by the physical delivery of the underlying asset in exchange for the (i) strike price K. Physical delivery occurs only when SN > K, in which case (i) the underlying asset can be sold at the price SN by the option holder, for a (i) (i) payoff SN − K. When SN > K, no delivery occurs and the payoff is 0, which (i) + is consistent with the expression C = SN − K . A similar procedure can be applied to other option contracts.
Vanilla options (examples)
Vanilla options are options whose claim payoff depends only on the terminal value ST of the underlying risky asset price at maturity time T . i) European options. The payoff of the European call option on the underlying asset no i with maturity N and strike price K is
(i) (i) SN − K if SN > K, (i) + C = SN − K = (i) 0 if SN < K. The moneyness at time t = 0, 1, ... , N of the European call option with strike price K on the asset no i is the ratio
(i) (i) S − K := t t = ... N Mt (i) , 0, 1, , . St
(i) The option is said to be “out of the money” (OTM) when Mt < 0,“in (i) the money” (ITM) when Mt > 0, and “at the money” (ATM) when (i) Mt = 0.
The payoff of the European put option on the underlying asset no i with exercise date N and strike price K is
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(i) (i) K − SN if SN 6 K, (i)+ C = K − SN = (i) 0 if SN > K. The moneyness at time t = 0, 1, ... , N of the European put option with strike price K on the asset no i is the ratio
(i) (i) K − S := t t = ... N Mt (i) , 0, 1, , . St ii) Binary options. Binary (or digital) options, also called cash-or-nothing options, are op- tions whose payoffs are of the form
(i) $1 if SN > K, 1 (i) C = [K,∞) SN = (i) 0 if SN < K, for binary call options, and
(i) $1 if SN 6 K, 1 (i) C = (−∞,K] SN = (i) 0 if SN > K, for binary put options.
iii) Collar and spread options. Collar and spread options provide other examples of vanilla options, whose payoffs can be constructed using call and put option payoffs, see, e.g., Exercises 3.11 and 3.12.
Exotic options (examples)
i) Asian options. The payoff of an Asian call option (also called option on average) on the underlying asset no i with exercise date N and strike price K is
N !+ 1 X (i) C = S − K . N + 1 t t=0 The payoff of an Asian put option on the underlying asset no i with exercise date N and strike price K is
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N !+ 1 X (i) C = K − S . N + 1 t t=0 We refer to Section 13.1 for the pricing of Asian options in continuous time. It can be shown, cf. Exercise 3.13 that Asian call option prices can be upper bounded by European call option prices.
Other examples of such options include weather derivatives (based on averaged temperatures) and volatility derivatives (based on averaged volatilities).
ii) Barrier options. The payoff of a down-an-out (or knock-out) barrier call option on the underlying asset no i with exercise date N, strike price K and barrier level B is (i) +1 C = SN − K n (i) o min St > B t=0,1,...,N (i) + (i) SN − K if min St > B, t=0,1,...,N = (i) 0 if min St 6 B. t=0,1,...,N
This option is also called a Callable Bull Contract with no residual value, or turbo warrant with no rebate, in which B denotes the call price B > K.
The payoff of an up-and-out barrier put option on the underlying asset no i with exercise date N, strike price K and barrier level B is
(i)+1 C = K − SN n (i) o Max St < B t=0,1,...,N (i)+ (i) K − SN if Max St < B, t=0,1,...,N = (i) 0 if Max St > B. t=0,1,...,N
This option is also called a Callable Bear Contract with no residual value, in which the call price B usually satisfies B 6 K. See Eriksson and Persson(2006) and Wong and Chan(2008) for the pricing of type R Callable Bull/Bear Contracts, or CBBCs, also called turbo warrants, which involve a rebate or residual value computed as the payoff of a down-and-in lookback option. We refer the reader to Chapters 11, 12 63 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault
and 13 for the pricing and hedging of related options in continuous time.
iii) Lookback options. The payoff of a floating strike lookback call option on the underlying asset no i with exercise date N is (i) (i) C = S − min St . N t=0,1,...,N
The payoff of a floating strike lookback put option on the underlying asset no i with exercise date N is (i) (i) C = Max St − S . t=0,1,...,N N
We refer to Section 10.4 for the pricing of lookback options in continuous time.
Options in insurance and investment
Such options are involved in the statements of Exercises 2.1 and 2.2.
Vanilla vs exotic options
(i) Vanilla options such as European or binary options, have a payoff φ(SN ) (i) that depends only on the terminal value SN of the underlying asset at ma- turity, as opposed to exotic or path-dependent options such as Asian, barrier, or lookback options, whose payoff may depend on the whole path of the un- derlying asset price until expiration time.
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Exotic vs Vanilla Options Vanilla options are called that way because: (A) They were first used for the trading of vanilla by the Maya beginning around the 14th century.
(B) “Plain vanilla” is the most standard and common of all ice cream flavors.
(C) To meet FDA standards, pure vanilla extract must contain 13.35 ounces of vanilla beans per gallon.
(D) Sir Charles C. Vanilla, FLS, was the early discoverer of the properties of Brownian motion in asset pricing.
Fig. 2.4: Take the Quiz.
2.4 Martingales and Conditional Expectations
Before proceeding to the definition of risk-neutral probability measures in dis- crete time we need to introduce more mathematical tools such as conditional expectations, filtrations, and martingales.
Conditional expectations
Clearly, the expected value of any risky asset or random variable is dependent on the amount of available information. For example, the expected return on a real estate investment typically depends on the location of this investment.
In the probabilistic framework the available information is formalized as a collection G of events, which may be smaller than the collection F of all avail- able events, i.e. G ⊂ F.∗
The notation E[F | G] represents the expected value of a random variable F given (or conditionally to) the information contained in G, and it is read “the conditional expectation of F given G”. In a certain sense, E[F | G] represents the best possible estimate of F in the mean-square sense, given the informa- tion contained in G.
The conditional expectation satisfies the following five properties, cf. Sec- tion 23.7 for details and proofs.
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i) E[FG | G] = GE[F | G] if G depends only on the information contained in G.
ii) E[G | G] = G when G depends only on the information contained in G.
iii) E[E[F | H] | G] = E[F | G] if G ⊂ H, called the tower property, cf. also Relation (23.38).
iv) E[F | G] = E[F ] when F “does not depend” on the information con- tained in G or, more precisely stated, when the random variable F is independent of the σ-algebra G.
v) If G depends only on G and F is independent of G, then
E[h(F , G) | G] = E[h(F , x)]x=G.
When H = {∅, Ω} is the trivial σ-algebra we have
E[F | H] = E[F ], F ∈ L1(Ω).
See (23.38) and (23.44) for illustrations of the tower property by conditioning with respect to discrete and continuous random variables.
Filtrations
The total amount of “information” available on the market at times t = 0, 1, ... , N is denoted by Ft. We assume that
Ft ⊂ Ft+1, t = 0, 1, ... , N − 1, which means that the amount of information available on the market in- creases over time.
(i) (i) (i) Usually, Ft corresponds to the knowledge of the values S0 , S1 , ... , St , i = 1, 2, ... , d, of the risky assets up to time t. In mathematical notation we (i) (i) (i) say that Ft is generated by S0 , S1 , ... , St , i = 1, 2, ... , d, and we usually write (i) (i) (i) Ft = σ S0 , S1 , ... , St , i = 1, 2, ... , d , t = 1, 2, ... , N, with F0 = {∅, Ω}.
Example: Consider the simple random walk
Zt := X1 + X2 + ··· + Xt, t > 0,
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where (Xt)t>1 is a sequence of independent, identically distributed {−1, 1} valued random variables. The filtration (or information flow) (Ft)t 0 gen- > erated by (Zt)t>0 is given by F0 = ∅, Ω , F1 = ∅, {X1 = 1}, {X1 =
−1}, Ω , and F2 = σ ∅, {X1 = 1, X2 = 1}, {X1 = 1, X2 = −1}, {X1 = −1, X2 = 1}, {X1 = −1, X2 = −1}, Ω .
The notation Ft is useful to represent a quantity of information available at time t. Note that different agents or traders may work with different filtra- tions. For example, an insider may have access to a filtration (Gt)t=0,1,...,N which is larger than the ordinary filtration (Ft)t=0,1,...,N available to an or- dinary agent, in the sense that
Ft ⊂ Gt, t = 0, 1, ... , N.
The notation E[F | Ft] represents the expected value of a random variable F given (or conditionally to) the information contained in Ft. Again, E[F | Ft] denotes the best possible estimate of F in mean-square sense, given the in- formation known up to time t.
We will assume that no information is available at time t = 0, which translates as E[F | F0] = E[F ] for any integrable random variable F . As above, the conditional expectation with respect to Ft satisfies the following five properties:
i) E[FG | Ft] = F E[G | Ft] if F depends only on the information con- tained in Ft.
ii) E[F | Ft] = F when F depends only on the information known at time t and contained in Ft.
iii) E[E[F | Ft+1] | Ft] = E[F | Ft] if Ft ⊂ Ft+1 (by the tower property, cf. also Relation (7.1) below).
iv) E[F | Ft] = E[F ] when F does not depend on the information contained in Ft.
v) If F depends only on Ft and G is independent of Ft, then
E[h(F , G) | Ft] = E[h(x, G)]x=F .
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Note that by the tower property (iii) the process t 7→ E[F | Ft] is a martin- gale, cf. e.g. Relation (7.1) for details.
Martingales
A martingale is a stochastic process whose value at time t + 1 can be esti- mated using conditional expectation given its value at time t. Recall that a stochastic process (Mt)t=0,1,...,N is said to be (Ft)t=0,1,...,N -adapted if the value of Mt depends only on the information available at time t in Ft, t = 0, 1, ... , N.
Definition 2.8. A stochastic process (Mt)t=0,1,...,N is called a discrete-time martingale with respect to the filtration (Ft)t=0,1,...,N if (Mt)t=0,1,...,N is (Ft)t=0,1,...,N -adapted and satisfies the property
E[Mt+1 | Ft] = Mt, t = 0, 1, ... , N − 1.
Note that the above definition implies that Mt ∈ Ft, t = 0, 1, ... , N. In other words, a random process (Mt)t=0,1,...,N is a martingale if the best possible prediction of Mt+1 in the mean-square sense given Ft is simply Mt.
In discrete-time finance, the martingale property can be used to character- ize risk-neutral probability measures, and for the computation of conditional expectations.
Exercise. Using the tower property (23.38) of conditional expectation, show that Definition 2.8 can be equivalently stated by saying that
E[Mn | Fk] = Mk, 0 6 k < n.
A particular property of martingales is that their expectation is constant over time.
Proposition 2.9. Let (Zn)n∈N be a martingale. We have
E[Zn] = E[Z0], n > 0. Proof. From the tower property (23.38) of conditional expectation, we have:
E[Zn+1] = E[E[Zn+1 | Fn]] = E[Zn], n > 0, hence by induction on n > 0 we have
E[Zn+1] = E[Zn] = E[Zn−1] = ··· = E[Z1] = E[Z0], n > 0.
Weather forecasting can be seen as an example of application of martingales. If Mt denotes the random temperature observed at time t, this process is a 68 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance martingale when the best possible forecast of tomorrow’s temperature Mt+1 given the information known up to time t is simply today’s temperature Mt, t = 0, 1, ... , N − 1.
Definition 2.10. A stochastic process (ξk)k>1 is said to be predictable if ξk depends only on the information in Fk−1, k > 1.
When F0 simply takes the form F0 = {∅, Ω} we find that ξ1 is a constant when (ξt)t=1,2,...,N is a predictable process. Recall that on the other hand, S(i) adapted S(i) the process t t=0,1,...,N is as t depends only on the information in Ft, t = 0, 1, ... , N, i = 1, 2, ... , d.
The discrete-time stochastic integral (2.9) will be interpreted as the sum of (1) (1) discounted profits and losses ξk Sek − Sek−1 , k = 1, 2, ... , t, in a portfolio (1) (1) holding a quantity ξk of a risky asset whose price variation is Sek − Sek−1 at time k = 1, 2, ... , t.
An important property of martingales is that the discrete-time stochastic integral (2.9) of a predictable process is itself a martingale, see also Propo- sition 7.1 for the continuous-time analog of the following proposition, which will be used in the proof of Theorem 3.5 below.∗
In the sequel, the martingale (2.9) will be interpreted as a discounted portfolio (1) (1) value, in which Sek − Sek−1 represents the increment in the discounted asset price and ξk is the amount invested in that asset, k = 1, 2 ... , N.
Theorem 2.11. Martingale transform. Given (Xk)k=0,1,...,N a martingale and (ξk)k=1,2,...,N a (bounded) predictable process, the discrete-time process (Mt)t=0,1,...,N defined by
t X Mt = ξk(Xk − Xk−1), t = 0, 1, ... , N, (2.9) k=1 | {z } Profit/loss is a martingale.
Proof. Given n > t > 0 we have " # n X E[Mn | Ft] = E ξk(Xk − Xk−1) Ft k=1 n X = E ξk(Xk − Xk−1) Ft k=1
∗ See here for a related discussion of martingale strategies in a particular case.
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t n X X = E [ξk(Xk − Xk−1) | Ft] + E [ξk(Xk − Xk−1) | Ft] k=1 k=t+1 t n X X = ξk(Xk − Xk−1) + E [ξk(Xk − Xk−1) | Ft] k=1 k=t+1 n X = Mt + E [ξk(Xk − Xk−1) | Ft] . k=t+1
In order to conclude to E [Mn | Ft] = Mt it suffices to show that
E [ξk(Xk − Xk−1) | Ft] = 0, t + 1 6 k 6 n.
First we note that when 0 6 t 6 k − 1 we have Ft ⊂ Fk−1, hence by the tower property (23.38) of conditional expectations we get
E [ξk(Xk − Xk−1) | Ft] = E [E [ξk(Xk − Xk−1) | Fk−1] | Ft] .
Next, since the process (ξk)k>1 is predictable, ξk depends only on the infor- mation in Fk−1, and using Property (ii) of conditional expectations we may pull out ξk out of the expectation since it behaves as a constant parameter given Fk−1, k = 1, 2, ... , n. This yields
E [ξk(Xk − Xk−1) | Fk−1] = ξkE [Xk − Xk−1 | Fk−1] = 0 (2.10) since
E [Xk − Xk−1 | Fk−1] = E [Xk | Fk−1] − E [Xk−1 | Fk−1] = E [Xk | Fk−1] − Xk−1 = 0, k = 1, 2, ... , N, because (Xk)k=0,1,...,N is a martingale. By (2.10), it follows that
E [ξk(Xk − Xk−1) | Ft] = E [E [ξk(Xk − Xk−1) | Fk−1] | Ft] = E [ξkE [Xk − Xk−1 | Fk−1] | Ft] = 0, for k = t + 1, ... , n.
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2.5 Market Completeness and Risk-Neutral Measures
As in the two time step model, the concept of risk-neutral probability mea- sure (or martingale measure) will be used to price financial claims under the absence of arbitrage hypothesis.∗ Definition 2.12. A probability measure P∗ on Ω is called a risk-neutral probability measure if under P∗, the expected return of each risky asset equals the return r of the riskless asset, that is
∗ (i) (i) E St+1 Ft = (1 + r)St , t = 0, 1, ... , N − 1, (2.11) i = 0, 1, ... , d. Here, E∗ denotes the expectation under P∗. (i) Since St ∈ Ft, denoting by
(i) (i) (i) S − S R := t+1 t t+1 (i) St the return of asset no i over the time interval (t, t + 1], t = 0, 1, ... , N − 1, Relation (2.11) can be rewritten as
" (i) (i) # ∗ (i) ∗ St+1 − St E R Ft = E Ft t+1 (i) St " (i) # ∗ St+1 = E Ft − 1 (i) St = r, t = 0, 1, ... , N − 1,
(i) (i) (i) o which means that the average of the return (St+1 − St )/St of asset n i under the risk-neutral probability measure P∗ is equal to the risk-free interest rate r.
In other words, taking risks under P∗ by buying the risky asset no i has a neutral effect, as the expected return is that of the riskless asset. The measure P] would yield a positive risk premium if we had
] (i) (i) E St+1 Ft = (1 + r˜)St , t = 0, 1, ... , N − 1, with r˜ > r, and a negative risk premium if r˜ < r.
In the next proposition we reformulate the definition of risk-neutral prob- ability measure using the notion of martingale.
∗ See also the Efficient Market Hypothesis.
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Proposition 2.13. A probability measure P∗ on Ω is a risk-neutral measure if and only if the discounted price process
(i) (i) S Se := t , t = 0, 1, ... , N, t (1 + r)t is a martingale under P∗, i.e.
∗ (i) (i) E Set+1 Ft = Set , t = 0, 1, ... , N − 1, (2.12) i = 0, 1, ... , d.
(i) t (i) Proof. It suffices to check that by the relation St = (1 + r) Set , Condi- tion (2.11) can be rewritten as
t+1 ∗ (i) t (i) (1 + r) E Set+1 Ft = (1 + r)(1 + r) Set , i = 1, 2, ... , d, which is clearly equivalent to (2.12) after division by (1 + r)t, t = 0, 1, ... , N − 1. Note that, as a consequence of Propositions 2.9 and 2.13, the discounted price (i) (i) t process Set := St /(1 + r) , t = 0, 1, ... , n, has constant expectation under the risk-neutral probability measure P∗, i.e.
∗ (i) (i) E Set = Se0 , t = 1, 2, ... , N, for i = 0, 1, ... , d.
In the sequel we will only consider probability measures P∗ that are equivalent to P, in the sense that they share the same events of zero probability. Definition 2.14. A probability measure P∗ on (Ω, F) is said to be equivalent to another probability measure P when
P∗(A) = 0 if and only if P(A) = 0, for all A ∈ F. (2.13)
Next, we restate in discrete time the first fundamental theorem of asset pric- ing, which can be used to check for the existence of arbitrage opportunities. Theorem 2.15. A market is without arbitrage opportunity if and only if it admits at least one equivalent risk-neutral probability measure. Proof. See Harrison and Kreps(1979) and Theorem 5.17 of Föllmer and Schied(2004). Next, we turn to the notion of market completeness, starting with the defi- nition of attainability for a contingent claim.
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Definition 2.16. A contingent claim with payoff C is said to be attainable (at time N) if there exists a (predictable) self-financing portfolio strategy ξ such that t t=1,2,...,N
d X (k) (k) • C = ξN SN = ξN SN , P − a.s. (2.14) k=0
ξ C In case t t=1,2,...,N is a portfolio that attains the claim payoff at time N i.e. ξ hedges , if (2.14) is satisfied, we also say that t t=1,2,...,N the claim payoff C. In case (2.14) is replaced by the condition
ξN • SN > C, we talk of super-hedging.
ξ C When a self-financing portfolio t t=1,2,...,N hedges a claim payoff , the arbitrage-free price πt(C) of the claim at time t is given by the value
πt(C) = ξt • St of the portfolio at time t = 0, 1, ... , N. Recall that arbitrage-free prices can be used to ensure that financial derivatives are “marked” at their fair value (mark to market). Note that at time t = N we have
πN (C) = ξN • SN = C, i.e. since exercise of the claim occurs at time N, the price πN (C) of the claim equals the value C of the payoff. Definition 2.17. A market model is said to be complete if every contingent claim is attainable. The next result can be viewed as the second fundamental theorem of asset pricing in discrete time. Theorem 2.18. A market model without arbitrage opportunities is complete if and only if it admits only one equivalent risk-neutral probability measure. Proof. See Harrison and Kreps(1979) and Theorem 5.38 of Föllmer and Schied(2004).
2.6 The Cox-Ross-Rubinstein (CRR) Market Model
We consider the discrete-time Cox-Ross-Rubinstein (Cox et al.(1979)) model, also called the binomial model, with N + 1 time instants t = 0, 1, ... , N and
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(0) d = 1 risky asset, see Sharpe(1978). In this setting, the price St of the riskless asset evolves as
(0) (0) t St = S0 (1 + r) , t = 0, 1, ... , N.
Let the return of the risky asset S(1) be defined as
(1) (1) St − S R := t−1 t = ... N t (1) , 1, 2, , . St−1
In the CRR model the return Rt is random and allowed to take only two values a and b at each time step, i.e.
Rt ∈ {a, b}, t = 1, 2, ... , N, with −1 < a < b and
P(Rt = a) > 0, P(Rt = b) > 0, t = 1, 2, ... , N.
(1) (1) That means, the evolution of St−1 to St is random and given by
(1) (1 + b)St−1 if Rt = b (1) (1) St = = (1 + Rt)St−1, t = 1, ... , N, (1) (1 + a)St−1 if Rt = a and t (1) (1) Y St = S0 (1 + Rk), t = 0, 1, ... , N. k=1 S(1) Note that the price process t t=0,1,...,N evolves on a binary recombining (or binomial) tree of the following type:∗
∗ Download the corresponding and that can be run here.
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2 S2 = S0(1 + b)
S1 = S0(1 + b)
S0 S2 = S0(1 + a)(1 + b)
S1 = S0(1 + a)
2 S2 = S0(1 + a) .
The discounted asset price is
(1) (1) S Se = t , t = 0, 1, ... , N, t (1 + r)t with
1 + b 1 S( ) R b et−1 if t = (1) 1 + r 1 + Rt (1) Se = = Se , t = 1, 2, ... , N, t 1 + r t−1 1 + a (1) Set−1 if Rt = a 1 + r and (1) t t (1) S0 Y (1) Y 1 + Rk Se = (1 + Rk) = Se . t (1 + r)t 0 1 + r k=1 k=1
23.84
19.07
15.26 9.54
12.21 7.63
9.77 6.1 3.81
7.81 4.88 3.05
6.25 3.91 2.44 1.53
5.0 3.12 1.95 1.22
4.0 2.5 1.56 0.98 0.61
2.0 1.25 0.78 0.49
1.0 0.62 0.39 0.24
0.5 0.31 0.2
0.25 0.16 0.1
0.12 0.08
0.06 0.04
0.03
0.02
Fig. 2.5: Discrete-time asset price tree in the CRR model.
In this model, the discounted value at time t of the portfolio is given by
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(0) (0) (1) (1) ξt • Xt = ξt Se0 + ξt Set , t = 1, 2, ... , N.
The information Ft known in the market up to time t is given by the (1) (1) (1) knowledge of S1 , S2 , ... , St , which is equivalent to the knowledge of (1) (1) (1) Se1 , Se2 , ... , Set or R1, R2, ... , Rt, i.e. we write
(1) (1) (1) (1) (1) (1) Ft = σ S1 , S2 , ... , St = σ Se1 , Se2 , ... , Set = σ(R1, R2, ... , Rt), t = 0, 1, ... , N, where, as a convention, S0 is a constant and F0 = {∅, Ω} contains no information.
Fig. 2.6: Discrete-time asset price graphs in the CRR model.
Theorem 2.19. The CRR model is without arbitrage opportunities if and only if a < r < b. In this case the market is complete and the equivalent risk-neutral probability measure P∗ is given by r − a b − r P∗(R = b | F ) = and P∗(R = a | F ) = , (2.15) t+1 t b − a t+1 t b − a t = 0, 1, ... , N − 1. In particular, (R1, R2, ... , RN ) forms a sequence of inde- pendent and identically distributed (i.i.d.) random variables under P∗, with
r − a b − r p∗ := P∗(R = b) = and q∗ := P∗(R = a) = , (2.16) t b − a t b − a t = 1, 2, ... , N. Proof. In order to check for arbitrage opportunities we may use Theorem 2.15 and look for a risk-neutral probability measure P∗. According to the definition of a risk-neutral measure this probability P∗ should satisfy Condition (2.11), i.e. ∗ (1) (1) E St+1 Ft = (1 + r)St , t = 0, 1, ... , N − 1.
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∗ (1) Rewriting E St+1 Ft as
∗ (1) ∗ (1) (1) E St+1 Ft = E St+1 St (1) ∗ (1) ∗ = (1 + a)St P (Rt+1 = a | Ft) + (1 + b)St P (Rt+1 = b | Ft), it follows that any risk-neutral probability measure P∗ should satisfy the equations
(1) ∗ (1) ∗ (1) (1 + b)St P (Rt+1 = b | Ft) + (1 + a)St P (Rt+1 = a | Ft) = (1 + r)St
∗ ∗ P (Rt+1 = b | Ft) + P (Rt+1 = a | Ft) = 1, i.e. ∗ ∗ bP (Rt+1 = b | Ft) + aP (Rt+1 = a | Ft) = r
∗ ∗ P (Rt+1 = b | Ft) + P (Rt+1 = a | Ft) = 1, with solution r − a b − r P∗(R = b | F ) = and P∗(R = a | F ) = , t+1 t b − a t+1 t b − a
∗ ∗ t = 0, 1, ... , N − 1. Since the values of P (Rt+1 = b | Ft) and P (Rt+1 = ∗ a | Ft) computed in (2.15) are non random, they are independent of the ∗ information contained in Ft up to time t. As a consequence, under P , the random variable Rt+1 is independent of R1, R2, ... , Rt, hence the sequence of random variables (Rt)t=0,1,...,N is made of mutually independent random variables under P∗, and by (2.15) we have r − a b − r P∗(R = b) = and P∗(R = a) = . t+1 b − a t+1 b − a Clearly, P∗ can be equivalent to P only if r − a > 0 and b − r > 0. In this case the solution P∗ of the problem is unique by construction, hence the market is complete by Theorem 2.18. As a consequence of Proposition 2.13, letting p∗ := (r − a)/(b − a), when N (1, 2, ... , n) ∈ {a, b} we have
∗ ∗ l ∗ N−l P (R1 = 1, R2 = 2, ... , RN = n) = (p ) (1 − p ) , where l, resp. N − l, denotes the number of times the term “b”, resp. “a”, N appears in the sequence (1, 2, ... , N ) ∈ {a, b} .
∗ The relation P(A | B) = P(A) is equivalent to the independence relation P(A ∩ B) = P(A)P(B) of the events A and B.
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Exercises
Exercise 2.1 Today I went to the Furong Peak mall. After exiting the Poon Way MTR station, I was met by a friendly investment consultant from NTRC Input, who recommended that I subscribe to the following investment plan. The plan requires to invest $ 2,550 per year over the first 10 years. No contribution is required from year 11 until year 20, and the total projected surrender value is $30,835 at maturity N = 20. The plan also includes a death benefit which is not considered here.
Surrender Value Year Total Premiums Guaranteed ($S) Projected at 3.25% Paid To-date ($S) Non-Guaranteed ($S) Total ($S) 1 2,550 0 0 0 2 5,100 2,460 140 2,600 3 7,650 4,240 240 4,480 4 10,200 6,040 366 6,406 5 12,750 8,500 518 9,018 10 25,499 19,440 1,735 21,175 15 25,499 22,240 3,787 26,027 20 25,499 24,000 6,835 30,835
Table 2.2: NTRC Input investment plan. a) Compute the constant interest rate over 20 years corresponding to this investment plan. b) Compute the projected value of the plan at the end of year 20, if the annual interest rate is r = 3.25% over 20 years. c) Compute the projected value of the plan at the end of year 20, if the annual interest rate r = 3.25% is paid only over the first 10 years. Does this recover the total projected value $30, 835?
Exercise 2.2 Today I went to the East mall. After exiting the Bukit Kecil MTR station, I was approached by a friendly investment consultant from Avenda Insurance, who recommended me to subscribe to the following in- vestment plan. The plan requires me to invest $ 3,581 per year over the first 10 years. No contribution is required from year 11 until year 20, and the total projected surrender value is $50,862 at maturity N = 20. The plan also includes a death benefit which is not considered here.
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Surrender Value Year Total Premiums Guaranteed ($S) Projected at 3.25% Paid To-date ($S) Non-Guaranteed ($S) Total ($S) 1 3,581 0 0 0 2 7,161 1,562 132 1,694 3 10,741 3,427 271 3,698 4 14,321 5,406 417 5,823 5 17,901 6,992 535 7,527 10 35,801 19,111 1,482 20,593 15 35,801 29,046 3,444 32,490 20 35,801 43,500 7,362 50,862
Table 2.3: Avenda Insurance investment plan. a) Using the following graph, compute the constant interest rate over 20 years corresponding to this investment.
16
15.5
15
14.5
14
13.5
13
12.5
12
11.5 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 x in % Fig. 2.7: Graph of the function x 7→ ((1 + x)21 − (1 + x)11)/x. b) Compute the projected value of the plan at the end of year 20, if the annual interest rate is r = 3.25% over 20 years. c) Compute the projected value of the plan at the end of year 20, if the annual interest rate r = 3.25% is paid only over the first 10 years. Does this recover the total projected value $50, 862?
Exercise 2.3 A lump sum of $100,000 is to be paid through constant yearly payments m at the end of each year over 10 years. a) Find the value of m. b) Assume that the amount remaining at the beginning of every year is in- vested at the interest rate r = 2%. How does this affect the value of the constant yearly payment m?
Exercise 2.4 Consider a two-step trinomial (or ternary) market model (St)t=0,1,2 with r = 0 and three possible return rates Rt ∈ {−1, 0, 1}. Show that the probability measure P∗ given by
∗ 1 ∗ 1 ∗ 1 P (Rt = −1) := , P (Rt = 0) := , P (Rt = 1) := 4 2 4 79 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault is risk-neutral.
(0) (0) Exercise 2.5 We consider a riskless asset valued Sk = S0 , k = 0, 1, ... , N, where the risk-free interest rate is r = 0, and a risky asset S(1) whose returns S(1) − S(1) R := k k−1 k = ... N k (1) , 1, 2, , , form a sequence of independent identi- Sk−1 cally distributed random variables taking three values {−b < 0 < b} at each time step, with
∗ ∗ ∗ ∗ ∗ ∗ p := P (Rk = b) > 0, θ := P (Rk = 0) > 0, q := P (Rk = −b) > 0, k = 1, 2, ... , N. The information known to the market up to time k is denoted by Fk. a) Determine all possible risk-neutral probability measures P∗ equivalent to P in terms of the parameter θ∗ ∈ (0, 1). b) Assume that the conditional variance
1 1 S( ) − S( ) ∗ k+1 k F = σ2 > k = ... N − Var (1) k 0, 0, 1, , 1, (2.17) Sk
of the asset return is constant and equal to σ2. Show that this condi- ∗ tion defines a unique risk-neutral probability measure Pσ under a certain ∗ condition on b and σ, and determine Pσ explicitly.
Exercise 2.6 We consider the discrete-time Cox-Ross-Rubinstein model with N + 1 time instants t = 0, 1, ... , N, with a riskless asset whose price πt t evolves as πt = π0(1 + r) , t = 0, 1, ... , N. The evolution of St−1 to St is given by (1 + b)St−1 St = (1 + a)St−1 with −1 < a < r < b. The return of the risky asset S is defined as
St − St−1 Rt := , t = 1, 2, ... , N, St−1 and Ft is generated by R1, R2, ... , Rt, t = 1, 2, ... , N. a) What are the possible values of Rt? b) Show that, under the probability measure P∗ defined by r − a b − r p∗ = P∗(R = b | F ) = , q∗ = P∗(R = a | F ) = , t+1 t b − a t+1 t b − a 80 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance
∗ t = 0, 1, ... , N − 1, the expected return E [Rt+1 | Ft] of S is equal to the return r of the riskless asset. ∗ c) Show that under P the process (St)t=0,1,...,N satisfies
∗ k E [St+k | Ft] = (1 + r) St, t = 0, 1, ... , N − k, k = 0, 1, ... , N.
Exercise 2.7 We consider the discrete-time Cox-Ross-Rubinstein model with N + 1 time instants t = 0, 1, ... , N, with a riskless asset whose price πt t evolves as πt = π0(1 + r) with r > 0, and a risky asset whose price St is given by t Y St = S0 (1 + Rk), t = 0, 1, ... , N, k=1 where the market return Rk are independent random variables taking two possible values a and b with −1 < a < r < b, and P∗ is the probability measure defined by r − a b − r p∗ = P∗(R = b | F ) = , q∗ = P∗(R = a | F ) = , t+1 t b − a t+1 t b − a t = 0, 1, ... , N − 1, where (Ft)t=0,1,...,N is the filtration generated by (Rt)t=1,2,...,N . ∗ ∗ a) Compute the conditional expected return E [Rt+1 | Ft] under P , t = 0, 1, ... , N − 1. b) Show that the discounted asset price process
S S t et t=0,1,...,N := πt t=0,1,...,N
∗ is a (nonnegative) (Ft)-martingale under P . ∗ Hint: Use the independence of market returns (Rt)t=1,2,...,N under P . ∗ β c) Compute the moment E [(SN ) ] for all β > 0. ∗ Hint: Use the independence of market returns (Rt)t=1,2,...,N under P . d) For any α > 0, find an upper bound for the probability
∗ P St > απt for some t ∈ {0, 1, ... , N} .
Hint: Use the fact that when (Mt)t=0,1,...,N is a nonnegative martingale we have β E[(MN ) ] P Max Mt > x 6 , x > 0, β > 1. (2.18) t=0,1,...,N xβ e) For any x > 0, find an upper bound for the probability
∗ P Max St > x . t=0,1,...,N 81 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault
Hint: Note that (2.18) remains valid for any nonnegative submartingale.
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